Exercise Sheet 1

MAT 182: Analysis für die Naturwissenschaften HS2017
Dr. C. Luchsinger
Exercise Sheet 1
The first exercise contains a repetition of the material covered in high school. In case
you don’t remember everything that’s no problem, the theory will be covered in the
lecture.
Hand in: Wednesday, 27.09.2017, ahead of the lecture.
Exercise 1 (5 points)
Consider the function y = f (x) = −x2 + 2x + 8.
a) (1 point) Find the roots of f (x).
b) (1 point) Find the vertex S of the parobola.
c) (2 points) Find the equation of the tangent at x = 2. If possible, avoid using
differential calculus.
d) (1 point) Sketch both the parabola and the tangent in a coordinate system.
Exercise 2 (6 points)
a) (2 points) Find all solutions of the equation
3x2 + 6x =
4
8
+
3 3x
Hint: use factorization
b) (2 points) Find the solution(s) of the equation 54x − 7 · 52x + 10 = 0.
Hint: use substitution
c) (2 points) Find the roots of x3 − 2x2 − 5x + 6.
Hint: guess one root and then apply polinomial division
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MAT 182: Analysis für die Naturwissenschaften HS2017
Dr. C. Luchsinger
Exercise 3 (4 points)
a) (2 points) Determine the
unknown variables x, y and z.
3x + 2y − z = 3 2x − y + z = 4 4x + 2y − 3z = −4 b) (2 points) Determine the
unknown variables x, y and z.
6x + y − 2z = 0 2x − y + z = 4 4x + 2y − 3z = −4 Hint: One equation is a linear combination of the two others.
Exercise 4 (6.5 points)
This exercise is meant to review the rules for exponential and logarithmic functions.
a) (1.5 points) Rewrite the following expressions using only one ln(.) expression:
1) 2 ln(a) − ln(3c)
1
ln(y)
3
1
3) 4 ln(m) − ln(n)
6
2) 3 ln(y) +
3
t
b) (1 point) Likewise for the exponential function: e5a · ex · (e1−t )
c) (2 points) Simplify whenever possible:
1) e− ln(2x)
2) e3 ln(w)
1
3) e 2 ln(2y)
1
4) e− 3 ln(d)
d) (1 point) Rearrange the following expressions whenever possible:
1) ln(a9 b)
2) ln(e3 + 5)
−x e
3) ln
b
2
e) (1 point) Solve 2x − 2x−1 = 2 for x.
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